The `6563 Å H_(2)` line emitted by hydrogen in a star is found to be red shifted by `15 Å`. Estimate the speed with which the star is receding from earth.

Options

A. `3.2xx10^(5)ms^(-1)`
B. `6.87xx10^(5)ms^(-1)`
C. `2xx10^(5)ms^(-1)`
D. `12.74xx10^(5)ms^(-1)`

Explanation

To solve the problem of estimating the speed with which the star is receding from Earth based on the redshift of the hydrogen line, we can follow these steps:

### Step 1: Identify the Given Values
We have the following values:
- Original wavelength of the hydrogen line, \( \lambda = 6563 \, \text{Å} \)
- Redshift amount, \( \Delta \lambda = 15 \, \text{Å} \)

### Step 2: Calculate the Shifted Wavelength
The shifted wavelength \( \lambda' \) can be calculated as:
\[ \lambda' = \lambda + \Delta \lambda = 6563 \, \text{Å} + 15 \, \text{Å} = 6578 \, \text{Å} \]

### Step 3: Use the Redshift Formula
The redshift \( z \) is defined as:
\[ z = \frac{\Delta \lambda}{\lambda} \]
Substituting the values we have:
\[ z = \frac{15 \, \text{Å}}{6563 \, \text{Å}} \]

### Step 4: Calculate the Redshift
Calculating the value of \( z \):
\[ z = \frac{15}{6563} \approx 0.002287 \]

### Step 5: Relate Redshift to Velocity
Using the approximation for non-relativistic speeds, the velocity \( v \) can be calculated using the formula:
\[ v = z \cdot c \]
where \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 6: Calculate the Velocity
Substituting the values:
\[ v = 0.002287 \cdot 3 \times 10^8 \, \text{m/s} \]
Calculating this gives:
\[ v \approx 686,000 \, \text{m/s} \quad \text{(or } 6.86 \times 10^5 \, \text{m/s)} \]

### Final Answer
The speed with which the star is receding from Earth is approximately \( 6.86 \times 10^5 \, \text{m/s} \).

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